A correction that is two functions

A counterpoise correction frozen at one reference geometry returns the uncorrected bond length exactly, at every reference — because a constant does not move a minimum. That leaves the case every real application is: the two atoms are not the same atom.

That whole argument is about a symmetric pair, where the correction splits in half and one number carries it. The splitting is easy to take for granted, and it has an answer that is not an approximation. The basis the other atom lent establishes that the borrowing is real; the question here is who does the borrowing.

The symmetric case is exact, and that is the problem #

Computing each half separately for a pair of identical centres, at forty-three separations from 0.8 to 9 bohr:

The two halves are equal at every one of them, to twelve decimal places.

That is not a numerical coincidence and it is not an approximation. The two centres are related by the reflection that swaps them, the whole calculation is invariant under it, and each half is the other’s mirror image. The share is exactly a half everywhere.

So the single frozen number is a complete description of a symmetric pair’s correction, and the completeness comes from the symmetry rather than from the correction being simple. The moment the symmetry goes, so does the completeness — and nothing about the symmetric calculation gives any warning of it.

The unequal case #

Take the same two three-function bases and put charges of 1 and 2 on the two centres — the one-electron analogue of a heteronuclear pair, where one atom is compact and the other diffuse.

Where a bond would be, essentially the whole correction is the heavy atom’s — the light centre’s share is under two per cent. By six bohr the two are equal, and beyond that the light centre’s is the larger.

They change places at 6.29 bohr, which is a number that can be located and which no version of the symmetric argument could have suggested exists.

Why the heavy atom borrows more #

The direction is worth explaining because the naive expectation goes the other way.

The diffuse atom borrows from the compact one is the usual sentence, and it is about which functions are useful — a diffuse atom’s tail is described by the compact atom’s functions badly, and vice versa. But the counterpoise half is not a measure of usefulness. It is how much that atom’s own energy improves when the ghosts arrive, and an energy improvement is measured against the atom’s own energy scale.

A centre of charge 2 has an exact energy of −2 hartree and a three-function basis leaves it further short in absolute terms than the same basis leaves a centre of charge 1, whose exact energy is −0.5. So the heavier centre has more room to improve, and any additional functions — even badly placed ones — recover more of it.

The asymmetry is in the energy scale rather than in the geometry — a distinction of the same kind a projector being unique while a basis is not draws, which is why it shrinks at large separation: once the ghosts are far away they help both centres by very little, and what is left is the geometrical part, which favours the diffuse atom.

That is the crossing at 6.29 bohr: the energy-scale effect dies faster than the geometrical one.

What it does to a frozen correction #

A frozen correction returns the uncorrected bond length exactly. The unequal split adds a second error to the frozen number, and it is of a different kind.

A frozen correction is wrong about the size of the correction away from its reference geometry — that was already known, and interpolation is the recommended remedy.

It is also wrong about the division, and this is new. The share moves from 0.4 per cent to 67 across the range computed, so a number frozen at a bonding geometry attributes essentially the whole correction to the heavy atom, and at a stretched geometry that attribution is backwards.

For a total energy that does not matter, since only the sum enters — which is why a symmetric pair can get away with one number and why a measurement a basis was not fitted to is the place such things surface. It matters wherever the correction is used per fragment — in a many-body decomposition, in an interaction energy between three fragments where the pairwise corrections are assembled, or in any scheme that assigns a basis-set error to an atom. A single frozen number cannot be decomposed, and decomposing it by assuming the symmetric answer is wrong by two orders of magnitude at short range.

What this says about interpolating the correction #

The standing proposal is to interpolate: compute the artefact at three geometries, fit a quadratic, and use it everywhere. That advice survives and gets sharper.

Interpolate each half separately, which costs nothing extra. The total is a sum of two functions of quite different shapes — one nearly flat and one falling by six orders of magnitude — and a quadratic through three points of the sum is fitting the sum of a slow curve and a fast one, which is exactly the case a low-order polynomial does worst on. Fitting the two separately is no more work, since both are computed anyway on the way to the total.

There is a further reason to prefer the split fit and it is about diagnosis rather than accuracy. Two curves show which centre is responsible for a large correction; their sum does not. A calculation with a surprisingly large superposition error is a calculation with a badly described atom in it somewhere, and the split says which one.

And the number of points needed differs between them. The heavy centre’s half falls smoothly and would be well caught by three; the light centre’s is not monotone at all in this calculation and would not be. Whether the non-monotonicity is real or is the solver’s floor is the one thing this model cannot settle, and it is a good reason to compute rather than to interpolate at all where the correction is small.

The number that is worth quoting #

Of everything above, one figure is the useful one to carry.

At a bonding separation the correction is 99.5 per cent one atom’s. Not sixty per cent, not two thirds — essentially all of it. A scheme that splits it evenly, which is what the symmetric intuition suggests, misattributes half the correction from one atom to the other, and half of a correction that is itself of the order of the binding energy is not a rounding.

The reason to quote that rather than the crossing at 6.29 bohr is that bonding separations are where calculations are done. The crossing is the more surprising number and the short-range split is the consequential one.

And the reason it is worth quoting at all is that nothing in a standard counterpoise calculation displays it. The two halves are computed — they have to be, since the correction is their sum — and then added, and the sum is what is reported. Two numbers are computed and one is printed, and the one that is thrown away is the one that says which atom the basis-set error belongs to.

What is quoted, and what is computed #

Nothing is quoted. The bases are even-tempered sets built for these calculations, the charges are 1 and 2, and every energy is a variational solution of a one-electron problem with closed-form integrals.

Each half is computed the way counterpoise defines it — the atom alone in its own basis, against the same atom with the partner’s functions present as ghosts — so the two halves are the same quantity computed twice with the roles exchanged, and their sum is the ordinary correction.

The symmetric control is the same code with one number changed. Both halves come back equal to twelve decimal places, which is the check that the asymmetry is the charges’ and not the routine’s.

Two atoms, three ways of being unlike #

The pair computed here differs in nuclear charge, which is one of three ways two centres can be unlike, and the other two are worth naming because the argument applies to them too and the sizes will differ.

Different charges, as here: the asymmetry is in the energy scale, it is largest at short range, and it reverses far out.

Different basis sizes on identical atoms — the unbalanced basis, which is much the commonest case in practice, since a calculation on a large molecule often carries a good basis on the atoms of interest and a poor one elsewhere. The poorly described atom has the most to gain from ghosts, so it takes almost the whole correction, and the ratio in these calculations runs into the thousands.

Different environments — two chemically identical atoms in different parts of a molecule. Nothing here reaches that, since the model has two centres and no environment, and it is the case where the asymmetry would be smallest and hardest to anticipate.

The common thread is that the split follows how badly each atom is described, not how large it is. That is a more useful rule than any of the numbers, and it is the one a reader can apply to a calculation far larger than any here.

The split exists because every function was given an owner #

One thing separates the two halves from the total, and it is worth stating because it decides how far the result travels.

The total correction needs only two descriptions of the pair: the one the molecule got and the one the separated fragments got. Whichever functions are present, the difference between those two is defined.

The halves need more. Computing one of them means running an atom with its partner’s functions present and its partner’s nucleus absent, and that requires knowing which functions were the partner’s. Every function here is centred on a nucleus, so the assignment is obvious and the division is exact.

It stops being obvious the moment a basis contains a function that belongs to neither. Bond-centred functions sit at the midpoint by design; floating functions are placed wherever they lower the energy and may end up anywhere; and for both there is no nucleus to attach them to and therefore no ghost calculation to define a half. The total is still perfectly well defined and the split is not.

So the asymmetry measured above is a property of the pair and of a convention — the convention that a basis is a set of functions owned by atoms. That convention is nearly universal and it is not forced, which is the standing caution of this whole anchor arriving in one more place.

What this cannot say #

Nothing here is a molecule. Two centres on a line with one electron between them is a model of the accounting, and the charges of 1 and 2 were chosen to make the asymmetry visible rather than to represent anything.

One electron, and no repulsion. The whole superposition question in real calculations is entangled with correlation, which recovers differently in a dimer basis than in a monomer one, and none of that is here. What is here is the one-electron part of the effect, which is the part that is unambiguous.

And the charges are 1 and 2, which is a more extreme pair than most bonds. A carbon–hydrogen pair is far less unequal and its asymmetry would be correspondingly smaller — the finding is that the split is not fixed, not that it is always this lopsided.

The non-monotone light half is at the solver’s floor. Its values run down to 10⁻⁹ hartree, and a variational energy computed to that precision is at the edge of what these closed-form integrals support; the shape of that curve should not be trusted in detail, only its size relative to the other.

And a basis is not a thing. A contraction is a decision made once and a superposition error is the price of it; neither is a property of the molecule, and both would be different for a different basis of the same size.

What that pair establishes is that the correction is not a number but a function of separation, and the next figure says which function.

What was checked #

A symmetric pair’s two halves are equal at every separation, to a part in $10^{12}$ — the control, and the statement that gives the asymmetry a scale.

An unequal pair’s halves differ by more than a factor of ten where a bond would be, checked across a range of separations rather than at one.

The lighter centre’s share runs from almost none of the correction to more than half of it, which is the finding.

And the two change places at a separation well beyond any bond, so the crossing is located rather than merely claimed to exist.

While the symmetric pair’s share does not move at all, checked separately, because a calculation that produced a moving share for both would have an error rather than a finding.

What the two halves are each a measure of #

The two numbers are not two instances of one quantity, and separating what each is about is what makes the asymmetry legible.

The heavy centre’s half measures how badly its own basis describes it. It is large because a three-function basis leaves a charge-2 centre far short in absolute energy, and any extra functions — however badly placed — recover some of that. It shrinks as the basis improves and it has nothing to do with the partner except that the partner is where the extra functions happen to be.

The light centre’s half measures how useful the partner’s functions are at that distance. It is small at short range because a compact partner’s functions are the wrong shape for a diffuse tail, and it grows relative to the other as the partner recedes and its functions start to look like the diffuse ones the light atom wanted.

So one half is about the basis and the other about the geometry, and they have different shapes because they are about different things. That is why no single function can stand in for the sum, and it is a stronger reason than the numbers alone give.

It is the same distinction a contraction is a decision made once draws between what a basis costs and what it buys, and the same one the basis the other atom lent drew between borrowing and being helped.

Still open: the three-fragment case #

The obvious open question is the three-fragment case, where the division stops being an accounting question. An interaction energy between three fragments is usually assembled from pairwise counterpoise corrections, and each pairwise correction is now known to be unequally divided — so the assembly double-counts one fragment’s share and undercounts another’s by amounts that do not cancel. Computing the three-body correction directly, and comparing it against the assembled pairwise one, would say how large that error is, and it is the first place where the division above has a consequence for a number anybody quotes.

The nearer question is the interpolation, now that it is two interpolations. Three computed values of each half and a quadratic through each would restore the derivative that freezing destroys, and the interesting output is how many points each half needs — the heavy centre’s is smooth and the light centre’s is not, so the answer is probably different numbers for the two, which is a more useful thing to tell somebody than a single count.

A third direction is the one the symmetric control makes available and nobody uses. Because a symmetric pair’s halves are exactly equal, any calculation that reports them unequal has a bug — in the ghost placement, in the basis assignment, or in which nuclei were switched off — and the check costs one extra energy. A projector is unique and a basis is not: the halves are basis-dependent quantities, and a symmetry they must respect is the cheapest test available on them.